
Then, by setting \(t=\log n\)  and using the Borel-Cantelli's lemma [1]}, the following holds almost surely:
\(\Vert P_n-P\Vert _{\mathcal {G}} \le 2\mathbb {E}\big [\Vert P_n^0\Vert _{\mathcal {G}} \big ]+ A_{n,\log n}+ \frac{2M \log n }{\sqrt{n}} .\) 
